Lectures 20 and 21 Propagation of electromagnetic waves in non-conducting materials

Introduction

In previous lectures we have studied electromagnetic waves travelling in vacuum. We now extend this treatment to propagation in non-conducting materials. In particular we will be interested in the propagation of waves from one media into a second different media (reflection and refraction effects).

Summary of electromagnetic waves in vacuum

Propagation in LIH non-conducting media

Propagation velocity and refractive index

The treatment in this case parallels that in a vacuum except that we must replace e₀ by e₀e_r and m₀ by m₀m_r. Hence the speed of light in the medium, c_m, is given by

c_m=1/(e₀e_rm₀m_r)^1/2 = c/(e_rm_r)^1/2

Where c is the speed of light in a vacuum.

However the only materials that have a m_r which is significantly different from 1 are non-LIH ones (e.g. iron). Hence for most LIH non-conducting materials c_m» c/(e_r)^1/2.

The refractive index n of a given material is defined as the speed of light in vacuum divided by that in the material. Hence

n = c/c_m = (e_rm_r)^1/2 » (e_r)^1/2 

or                     n² » e_r

Because e_r always has a value greater than one, the speed of light in a material is always less than in a vacuum.

As both e_r and n may vary strongly with frequency, discrepancies may arise when comparing values of e_r (often the static, DC value is used) with n² (generally the value appropriate to optical frequencies).

Relationship between E and B

In vacuum E=cB.

In a material this is modified to

E=c_mB=cB/n

and because

H=B/(m₀m_r)»B/m₀ as m_r»1

Þ H=nE/(cm₀)

Reflection and refraction at the interface between two different materials

The aim is to establish the properties of electromagnetic waves when they encounter a plane interface separating two different non-conducting materials (generally two different dielectrics). In particular equations will be derived which give the fractions of the incident wave which are reflected and transmitted at the interface

Mathematical description of a plane wave not propagating along one of the principal axes

So far we have only considered waves which propagate along one of the principal axes. For example a plane wave propagating along the x-axis is described by an equation of the form E=E₀sin(kx-wt), one propagating along the y-axis by E=E₀sin(ky-wt) etc.

In the following we will need to consider plane waves which do not propagate along one of these principal axes. However we can always resolve the direction of the wave onto two or more of the principal axes.

For example in the figure below the wave lies in the x/y-plane and propagates in a direction which makes an angle q to the y-axis. There are hence components cosq along the y-axis and sinq along the x-axis. The equation for this plane wave is hence of the form

E=E₀sin(k(xsinq+ycosq)-wt)

The wave vector, k, in the above equations is given by 2p/l where l is the wavelength of the wave. In addition if the wave propagates in a material with a velocity c_m then

w=c_mk=(c/n)k

where w is the angular frequency of the wave. Hence

k=nw/c

Boundary conditions for E, D, B and H at an interface

In the lectures on dielectrics and magnetic materials it was shown that, in the absence of free surface charge and conduction currents, the following boundary conditions existed for E, D, B and H

D and B: Normal components continuous

E and H: Tangential components continuous

These boundary conditions will be used below for electromagnetic waves incident at the boundary between two materials.

Frequency and direction of reflected and refracted waves

In the figure below a plane electromagnetic wave travelling in a material of refractive index n₁ is incident on the plane boundary with a second material of refractive index n₂. The incident wave propagates at an angle q_i to the normal of the boundary and has E- and H-fields of magnitude E_i and H_i. The E-field of the incident wave lies in the plane of incidence (the x/y-plane), this is known as the E parallel configuration.

For this configuration the B- and H-fields must be normal to the plane of incidence.

In general there will be, in addition to the incident wave, a reflected wave at an angle q_r and a refracted or transmitted wave at an angle q_t. Each of these waves will have E- and H-fields as shown with amplitudes related by the expression H=nE/m₀c where n is the refractive index of the appropriate material.

The E-fields of the three waves are given by the following expressions

Where E_i0, E_r0 and E_t0 are the amplitudes of the E-fields and k₁ and k₂ and w₁ and w₂ are the wavevectors and angular frequencies of the waves in the two materials.

The boundary condition for E requires that the tangential component (the component parallel to the boundary) be continuous. Hence if we evaluate this component on both sides of the boundary we must obtain the same result.

The tangential component of each E-field is given by the magnitude of the E-field multiplied by cosq, where q is the appropriate angle.

On the side of the boundary in material 1 the total tangential component of the E-field is the difference of the components due to the incident and reflected waves (see above figure). In material 2 there is only the transmitted wave.

Hence the requirement that the tangential component of the E-field be continuous can be written 

E_icosq_iç_y=0 - E_rcosq_rç_y=0=E_tcosq_tç_y=0                    (D)

Where ç_y=0 indicates that the preceding expression is evaluated at the boundary y=0.

Substituting in (D) the expressions for E_i, E_r and E_t given by (A), (B) and (C) and evaluated for y=0

E_i0cosq_isin(k₁xsinq_i-w₁t)-E_r0cosq_rsin(k₁xsinq_r-w₁t)=E_t0cosq_tsin(k₂xsinq_t-w₂t)                 (E)

This equation must hold at all times and for all values of x. This is only possible if all the coefficients of x and all the coefficients of t are equal. This requires

w₁=w₂ and k₁sinq_i=k₁sinq_r=k₂sinq_t

Hence

The above results would have been obtained if E were polarized normal to the plane of incidence (H polarised parallel to the plane of incidence). These results hence apply to any incident wave.

Amplitudes of the reflected and refracted waves

The above procedure provided information on the frequencies and directions of the incident, reflected and transmitted waves. The next step is to derive expressions which give their relative amplitudes.

Returning to equation (E) above, which is valid when E is parallel to the plane of incidence, the spatial and time dependent components have been shown to be equal. Hence (E) reduces to

E_i0cosq_i - E_r0cosq_i=E_t0cosq_t             (F)

Where q_r has been replaced by q_i.

In addition the tangential component of the H-field must be continuous at the boundary between the two materials. For the present case H is normal to the incident plane so that the tangential component of H is simply H.

For the tangential component of H to be continuous we must therefore have

H_i0+H_r0=H_t0

But H=nE/m₀c so the above can be written as

n₁E_i0+n₁E_r0=n₂E_t0                  (G)

Equations (F) and (G) can now be used to eliminate either E_r0 or E_t0.

Eliminating E_t0:

From (G) E_t0=(E_i0+E_r0)n₁/n₂

Substituting into (F)

E_i0cosq_i-E_r0cosq_i=(E_i0+E_r0)(n₁/n₂)cosq_t

E_i0(cosq_i-(n₁/n₂)cosq_t)=E_r0(cosq_i+(n₁/n₂)cosq_t)

Multiplying through by n₂

E_i0(n₂cosq_i-n₁cosq_t)=E_r0(n₂cosq_i+n₁cosq_t)

                              (H)

Now eliminating E_r0. From (G)

E_r0=(n₂E_t0-n₁E_i0)/n₁

Substituting into (F)

E_i0cosq_i-(n₂E_t0-n₁E_i0)/n₁cosq_i=E_t0cosq_t

E_i0n₁cosq_i-(n₂E_t0-n₁E_i0)cosq_i=E_t0n₁cosq_t

2E_i0n₁cosq_i=E_t0(n₁cosq_t+n₂cosq_i)

                    (I)

r_// and t_//, which relate the amplitudes of the reflected and transmitted E-fields to that of the incident E-field, are known as the reflection and transmission coefficients for E parallel to the plane of incidence.

The polarisation of E can also be aligned normal to the plane of incidence (H is now parallel to the plane of incidence). The boundary conditions that the tangential components of E and H are continuous now require:

E_i0+E_r0=E_t0

H_i0cosq_i-H_r0cosq_i=H_t0cosq_t

Again eliminating either E_r0 or E_t0 from these two equations leads to expressions for the E perpendicular reflection r_^ and transmission t_^ coefficients:

                               (J)

                              (K)

The equation (H)-(K) are known as the Fresnel relationships or equations.

 The signs of these equations give the relative phases of the waves. If positive there is no phase change, if negative there is a p phase change.

Properties of the Fresnel Equations

The properties can be more easily seen if we consider a special case where one of the materials is air (n=1) and the other has a refractive index n.

Consider the case where the wave is incident from air. Hence n₁=1 and n₂=n. The Fresnel equations become

with sinq_i/sinq_t=n

The above figure (plotted for n=2) shows a number of points:

For normal incidence (q_i=q_t=0) both r_// and r_^ and t_// and t_^ have the same magnitudes

êr_//ê=êr_^ê=(n-1)/(n+1), êt_//ê=êt_^ê=2/(1+n)

As q_i®90° the magnitudes of the reflection coefficients tend to 1 (total reflection) and the magnitude of the transmission coefficients tend to zero (zero transmission).

For a certain angle q_B, known as the Brewster angle, r_// becomes zero whereas r_^ remains non-zero. For this angle light can only be reflected with E perpendicular to the plane of incidence.

If unpolarised light is incident at the Brewster angle then the reflected light will be polarised.

The Brewster angle is found by setting r_//=0.

Þ        ncosq_i=cosq_t              (L)

In addition Snell’s law gives us

nsinq_t=sinq_i                            (M)

dividing (L) by (M)

(ncosq_i)/(nsinq_t)=cosq_i/sinq_t=cosq_t/sinq_i    

Þ cosq_isinq_i= cosq_tsinq_t

and hence

sin2q_i=sin2q_t as 2sinqcosq=sin2q

The equality sin2q_i=sin2q_t implies that either q_i=q_t or q_i=90-q_t.  The former can not be correct as we also must have nsinq_t=sinq_i and n¹1.

Hence at the Brewster angle we must have q_i=90-q_t and Snell’s law becomes

nsinq_t=sinq_B   nsin(90-q_B)=sinq_B

but sin(90-q)=cosq hence

ncosq_B=sinq_B             Þ        n=sinq_B/cosq_B=tanq_B

Waves propagating from a material into air

We now have n₁=n and n₂=1. Fresnel’s equations have the form

and nsinq_i=sinq_t

In this case the equations for r_// and r_^ and for t_// and t_^ are interchanged compared to those for when the wave is incident from air.

However because nsinq_i=sinq_t Þq_t>q_i. For q_i equal to a certain value, the critical angle q_c, sinq_t becomes equal to unity and hence q_t=90°.

At this point both reflection coefficients become equal to one and the waves are totally reflected. As the value of sinq_t can not exceed unity, for q_i>q_c the reflection coefficients remain equal to unity.

Hence for q_i>q_c all the incident light is reflected. This is known as ‘total internal reflection’.

q_i=q_c occurs when sinq_t=1. Hence  nsinq_c=1 Þ sinq_c=1/n

Power reflection and transmission coefficients

The reflection and transmission coefficients derived above give the amplitudes of the electric fields associated with the waves.

However in general it is the power of the wave which is measured experimentally. From Lecture 19 the power density of an electromagnetic wave is given by the Poynting vector N=ExH.

This has a magnitude EH = nE²/(cm₀), using H=nE/(cm₀).

Hence the energy density µnE²

The coefficients which give the fraction of energy reflected or transmitted at a boundary between two materials equal the appropriate values of r² or t² with the inclusion of the appropriate value(s) of n.

If R is the power reflection coefficient at an interface between a material of refractive index n₁ and one of n₂ then

where r is the appropriate reflection coefficient given by the Fresnel relationships.

Similarly if T is the power transmission coefficient

because energy must always be conserved

R+T=1

Example:

What are the reflection and transmission power coefficients for light incident normally from air into a material of refractive index n?

q_i=q_t=0° and ½r_//½=½r_^½=(n-1)/(n+1)

  Hence  

  ½t_//½=½t_^½=2/(n+1)

Conclusions

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